

\((V,\in )\)  models \((*)\text{-}\mathcal {A}\) ;


\(\mathbf {NS}_{\omega _1}\)  is precipitousSee [1]} for a definition of precipitousness and a discussion of its properties. A key observation is that \(\mathbf {NS}_{\omega _1}\)  being precipitous is independent of \(\mathsf {CH}\)  (see for example [1]}), while \((*)\text{-}\mathcal {A}\)  entails \(2^{\aleph _0}=\aleph _2\)  (for example by the results of [3]}).
Another key point is that we stick to the formulation of \(\mathbb {P}_{\mathrm {max}}\)  as in [3]} so to
be able in its proof to quote verbatim from [3]} all the relevant results on \(\mathbb {P}_{\mathrm {max}}\) -preconditions we will use.
It is however possible to develop \(\mathbb {P}_{\mathrm {max}}\)  focusing on Woodin's countable tower rather than
on the precipitousness of \(\mathbf {NS}_{\omega _1}\)  to define the notion of \(\mathbb {P}_{\mathrm {max}}\) -precondition.
Following this approach in
all its scopes, one should be able to reformulate Thm. REF (REF )
omitting the request that
\(\mathbf {NS}_{\omega _1}\)  is precipitous. We do not explore this venue any further.
 in \(V\)  and
the \(\in _{\mathbf {NS}_{\omega _1}}\cup \mathcal {A}^V\) -theoryE.g. we regard \(A\subseteq (2^\omega )^k\)  as a \(k\) -ary predicate symbol for any \(A\in \mathcal {A}\) . of \(V\)  has as model companion the
\(\in _{\mathbf {NS}_{\omega _1}}\cup \mathcal {A}^V\) -theory of \(H_{\omega _2}\) .


